Equitable Division of a Path

TL;DR

This paper investigates fair and efficient resource allocation on a path, demonstrating computational hardness for combined fairness and efficiency, and providing polynomial-time algorithms under relaxed conditions or specific valuation structures.

Contribution

It introduces algorithms for connected EQ1 allocations with relaxed efficiency constraints and extends results to various valuation types and fairness notions.

Findings

Achieving EQ1 with efficiency measures is computationally hard for binary additive valuations.

Connected EQ1 allocations can be computed in polynomial time with a given agent order.

Tractability is possible for binary additive valuations with interval structure when efficiency is required.

Abstract

We study fair resource allocation under a connectedness constraint wherein a set of indivisible items are arranged on a path and only connected subsets of items may be allocated to the agents. An allocation is deemed fair if it satisfies equitability up to one good (EQ1), which requires that agents' utilities are approximately equal. We show that achieving EQ1 in conjunction with well-studied measures of economic efficiency (such as Pareto optimality, non-wastefulness, maximum egalitarian or utilitarian welfare) is computationally hard even for binary additive valuations. On the algorithmic side, we show that by relaxing the efficiency requirement, a connected EQ1 allocation can be computed in polynomial time for any given ordering of agents, even for general monotone valuations. Interestingly, the allocation computed by our algorithm has the highest egalitarian welfare among all allocations consistent with the given ordering. On the other hand, if efficiency is required, then tractability can still be achieved for binary additive valuations with interval structure. On our way, we strengthen some of the existing results in the literature for other fairness notions such as envy-freeness up to one good (EF1), and also provide novel results for negatively-valued items or chores.